# Differential geometry III - Riemannian geometry

## Summary

This course will serve as a first introduction to the geometry of Riemannian manifolds, which form an indispensible tool in the modern fields of differential geometry, analysis and theoretical physics.

## Content

Differentiable manifolds appearing in fields ranging from PDEs to theoretical physics usually come equipped with a Riemannian metric; this is simply a symmetric tensor defining an inner product on the space of tangent vectors over each point of the manifold. In contrast to a simple differentiable manifold which only carries topological information, a Riemannian manifold also contains a geometric structure: A Riemannian metric allows us to define notions such as the length of a curve or the distance between two points in the manifold.

In this course, we will introduce the basic concepts associated to Riemannian geometry, such as the Riemannian metric, the curvature tensor and the notion of a geodesic curve. We will then proceed to explore the geometric properties of these objects, in many cases extending ideas and results from Euclidean geometry to this more general setting. We will also discuss the consequences of certain geometric assumptions on the topology of Riemannian manifolds.

The course will cover the following topics:

- Riemannian metrics
- Riemannian connections and geodesics
- The curvature tensors
- The metric structure of a Riemannian manifold and the Hopf-Rinow theorem
- The geometry of hypersurfaces
- Comparison theorems and topological applications

## Keywords

Differential geometry; Riemannian metric; Curvature tensor; geodesics

## Learning Prerequisites

## Required courses

Differential Geometry II - Smooth manifolds, Analysis I-IV.

## Important concepts to start the course

Differentiable manifold, tangent-cotangent space, vector fields.

## Learning Outcomes

By the end of the course, the student must be able to:

- Define the central objects of Riemannian geometry (Riemannian metric, geodesics, etc)
- Use these objects together with the fundamental identities satisfied by them to solve problems.
- Prove the main theorems appearing in the course.

## Transversal skills

- Assess progress against the plan, and adapt the plan as appropriate.
- Demonstrate a capacity for creativity.
- Demonstrate the capacity for critical thinking
- Access and evaluate appropriate sources of information.

## Teaching methods

2h lectures + 2h exercises

## Assessment methods

Final exam.

Dans le cas de l'art. 3 al. 5 du Règlement de section, l'enseignant décide de la forme de l'examen qu'il communique aux étudiants concernés.

## Supervision

Office hours | No |

Assistants | Yes |

Forum | No |

## Resources

## Virtual desktop infrastructure (VDI)

No

## Bibliography

There are many introductory books on Riemannian geometry, unfortunately most of them intended for an audience of graduate students. We will follow closely the exposition of the following book:

do Carmo, Manfredo; Riemannian geometry. Birkhäuser Boston, Inc., Boston, MA, 1992.

We will sometimes use material from:

Petersen, Peter; Riemannian geometry. Springer-Verlag New York 2006.

Both manuscripts are available at the EPFL library.

**Ressources en bibliothèque**

## Ressources en bibliothèque

## Moodle Link

## Prerequisite for

Differential Geometry IV - General Relativity

## In the programs

**Semester:**Spring**Exam form:**Written (summer session)**Subject examined:**Differential geometry III - Riemannian geometry**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks**Type:**optional

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